15. Let *H_k(n)* be the number of vectors *x₁*,...,*x_k* for which each *x_i* is a positive integer satisfying 1 ≤ *x_i* ≤ *n* and *x₁* ≤ *x₂* ≤ ... ≤ *x_k*.

(a) Without any computations, argue that

*H₁(n)* = *n*

*H_k(n)* = ∑_*j*=1^*n* *H_k-1(j)*

*k* > 1 HINT: How many vectors are there in which *x_k* = *j*?

(b) Use the preceding recursion to compute *H₃(5)*.

HINT: First compute *H₂(n)* for *n* = 1, 2, 3, 4, 5.